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real reciprocal function
Contents
Context
Analysis
Algebra
- algebra, higher algebra
- universal algebra
- monoid, semigroup, quasigroup
- nonassociative algebra
- associative unital algebra
- commutative algebra
- Lie algebra, Jordan algebra
- Leibniz algebra, pre-Lie algebra
- Poisson algebra, Frobenius algebra
- lattice, frame, quantale
- Boolean ring, Heyting algebra
- commutator, center
- monad, comonad
- distributive law
Group theory
Ring theory
Module theory
Contents
Definition
Implicit definition
In real analysis, the reciprocal is a partial function implicitly defined over the non-zero real numbers by the equation . This is the definition commonly used when defining the real numbers as a field.
By the exponential and natural logarithm functions
The reciprocal is piecewise defined as
This definition implies that the reciprocal is analytic in each of the two connected components of the domain.
By power series
Let us define the functions and from the open subinterval of the real numbers to the real numbers as the locally convergent power series
The reciprocal is then piecewise defined as
Equivalently, given an element , the reciprocal is then piecewise defined as
This definition implies that the reciprocal is analytic in each of the two connected components of the domain.
Last revised on August 21, 2024 at 01:54:54.
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